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Applied Mathematics and Computation, Volume 236, 1 June 2014, Pages 184-194.
Tiberiu Harko, Francisco S.N. Lobo, M.K. Mak .
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom and
Centro de Astronomia e Astrofísica da Universidade de Lisboa, Campo Grande, Edificío C8, 1749-016 Lisboa, Portugal and
Department of Computing and Information Management, Hong Kong Institute of Vocational Education, Chai Wan, Hong Kong, PR China.
Abstract
In this paper, the exact analytical solution of the Susceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametric form. By using the exact solution we investigate some explicit models corresponding to fixed values of the parameters, and show that the numerical solution reproduces exactly the analytical solution. We also show that the generalization of the SIR model, including births and deaths, described by a nonlinear system of differential equations, can be reduced to an Abel type equation. The reduction of the complex SIR model with vital dynamics to an Abel type equation can greatly simplify the analysis of its properties. The general solution of the Abel equation is obtained by using a perturbative approach, in a power series form, and it is shown that the general solution of the SIR model with vital dynamics can be represented in an exact parametric form.
Additional Information:
The mathematical study of the outbreak and spread of diseases has been a major research area for many years. A simple deterministic (compartmental) model predicting the behavior of epidemic outbreaks was formulated by A. G. McKendrick and W. O Kermack in 1927 [1]. This mathematical three compartment epidemic model, called the Susceptible-Infected-Recovered (SIR) model, is described by a nonlinear system of ordinary differential equations. The compartments used for this model consist of three classes, representing the number of individuals not yet infected with disease, the number of infected individuals, who are capable of spreading the disease to those in the susceptible category, and of the individuals who have been infected and then recovered from the disease, respectively. We have obtained the exact analytical solution of the SIR epidemic model in a parametric form. The solution describes exactly the dynamical evolution of the SIR system for any given initial conditions, and for arbitrary values of the model parameters. By using the exact solution some explicit models corresponding to fixed values of the parameters have been investigated, and we have shown that the analytical solution reproduces exactly the numerical solution. Any change in the numerical values of the initial conditions and/or of the rate parameters does not affect the validity of the solution. The numerical values of the two integration constants in the solution are determined by the model parameters and the initial conditions. The generalization of the SIR model, including births and deaths, described by a strongly nonlinear system of differential equations, can be reduced to an Abel type equation. This Abel equation can be studied by means of semi-analytical/numerical methods, thus leading to a significant simplification in the study of the model. Once the general solution of the Abel equation is known, the general solution of the SIR epidemic model with deaths can be obtained in an exact parametric form. The comparison of the exact numerical solution for different order approximations obtained by iteratively solving the Abel and the numerical solution is also performed. After twenty steps the iterative and the numerical solution approximately overlap.
Finding exact solutions of mathematical models is important because biologists could use it to design and run experiments to observe the spread of infectious diseases by introducing natural initial conditions. Through these experiments, as well as through mathematical modelling, one can learn the ways on how to control the spread of epidemics.
[1] W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond \textbf{A 115},700-721 (1927).
